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diff --git a/src/main/java/org/apache/commons/math/geometry/Rotation.java b/src/main/java/org/apache/commons/math/geometry/Rotation.java new file mode 100644 index 0000000..ee3f4b7 --- /dev/null +++ b/src/main/java/org/apache/commons/math/geometry/Rotation.java @@ -0,0 +1,1072 @@ +/* + * Licensed to the Apache Software Foundation (ASF) under one or more + * contributor license agreements. See the NOTICE file distributed with + * this work for additional information regarding copyright ownership. + * The ASF licenses this file to You under the Apache License, Version 2.0 + * (the "License"); you may not use this file except in compliance with + * the License. You may obtain a copy of the License at + * + * http://www.apache.org/licenses/LICENSE-2.0 + * + * Unless required by applicable law or agreed to in writing, software + * distributed under the License is distributed on an "AS IS" BASIS, + * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + * See the License for the specific language governing permissions and + * limitations under the License. + */ + +package org.apache.commons.math.geometry; + +import java.io.Serializable; + +import org.apache.commons.math.MathRuntimeException; +import org.apache.commons.math.exception.util.LocalizedFormats; +import org.apache.commons.math.util.FastMath; + +/** + * This class implements rotations in a three-dimensional space. + * + * <p>Rotations can be represented by several different mathematical + * entities (matrices, axe and angle, Cardan or Euler angles, + * quaternions). This class presents an higher level abstraction, more + * user-oriented and hiding this implementation details. Well, for the + * curious, we use quaternions for the internal representation. The + * user can build a rotation from any of these representations, and + * any of these representations can be retrieved from a + * <code>Rotation</code> instance (see the various constructors and + * getters). In addition, a rotation can also be built implicitly + * from a set of vectors and their image.</p> + * <p>This implies that this class can be used to convert from one + * representation to another one. For example, converting a rotation + * matrix into a set of Cardan angles from can be done using the + * following single line of code:</p> + * <pre> + * double[] angles = new Rotation(matrix, 1.0e-10).getAngles(RotationOrder.XYZ); + * </pre> + * <p>Focus is oriented on what a rotation <em>do</em> rather than on its + * underlying representation. Once it has been built, and regardless of its + * internal representation, a rotation is an <em>operator</em> which basically + * transforms three dimensional {@link Vector3D vectors} into other three + * dimensional {@link Vector3D vectors}. Depending on the application, the + * meaning of these vectors may vary and the semantics of the rotation also.</p> + * <p>For example in an spacecraft attitude simulation tool, users will often + * consider the vectors are fixed (say the Earth direction for example) and the + * frames change. The rotation transforms the coordinates of the vector in inertial + * frame into the coordinates of the same vector in satellite frame. In this + * case, the rotation implicitly defines the relation between the two frames.</p> + * <p>Another example could be a telescope control application, where the rotation + * would transform the sighting direction at rest into the desired observing + * direction when the telescope is pointed towards an object of interest. In this + * case the rotation transforms the direction at rest in a topocentric frame + * into the sighting direction in the same topocentric frame. This implies in this + * case the frame is fixed and the vector moves.</p> + * <p>In many case, both approaches will be combined. In our telescope example, + * we will probably also need to transform the observing direction in the topocentric + * frame into the observing direction in inertial frame taking into account the observatory + * location and the Earth rotation, which would essentially be an application of the + * first approach.</p> + * + * <p>These examples show that a rotation is what the user wants it to be. This + * class does not push the user towards one specific definition and hence does not + * provide methods like <code>projectVectorIntoDestinationFrame</code> or + * <code>computeTransformedDirection</code>. It provides simpler and more generic + * methods: {@link #applyTo(Vector3D) applyTo(Vector3D)} and {@link + * #applyInverseTo(Vector3D) applyInverseTo(Vector3D)}.</p> + * + * <p>Since a rotation is basically a vectorial operator, several rotations can be + * composed together and the composite operation <code>r = r<sub>1</sub> o + * r<sub>2</sub></code> (which means that for each vector <code>u</code>, + * <code>r(u) = r<sub>1</sub>(r<sub>2</sub>(u))</code>) is also a rotation. Hence + * we can consider that in addition to vectors, a rotation can be applied to other + * rotations as well (or to itself). With our previous notations, we would say we + * can apply <code>r<sub>1</sub></code> to <code>r<sub>2</sub></code> and the result + * we get is <code>r = r<sub>1</sub> o r<sub>2</sub></code>. For this purpose, the + * class provides the methods: {@link #applyTo(Rotation) applyTo(Rotation)} and + * {@link #applyInverseTo(Rotation) applyInverseTo(Rotation)}.</p> + * + * <p>Rotations are guaranteed to be immutable objects.</p> + * + * @version $Revision: 1067500 $ $Date: 2011-02-05 21:11:30 +0100 (sam. 05 févr. 2011) $ + * @see Vector3D + * @see RotationOrder + * @since 1.2 + */ + +public class Rotation implements Serializable { + + /** Identity rotation. */ + public static final Rotation IDENTITY = new Rotation(1.0, 0.0, 0.0, 0.0, false); + + /** Serializable version identifier */ + private static final long serialVersionUID = -2153622329907944313L; + + /** Scalar coordinate of the quaternion. */ + private final double q0; + + /** First coordinate of the vectorial part of the quaternion. */ + private final double q1; + + /** Second coordinate of the vectorial part of the quaternion. */ + private final double q2; + + /** Third coordinate of the vectorial part of the quaternion. */ + private final double q3; + + /** Build a rotation from the quaternion coordinates. + * <p>A rotation can be built from a <em>normalized</em> quaternion, + * i.e. a quaternion for which q<sub>0</sub><sup>2</sup> + + * q<sub>1</sub><sup>2</sup> + q<sub>2</sub><sup>2</sup> + + * q<sub>3</sub><sup>2</sup> = 1. If the quaternion is not normalized, + * the constructor can normalize it in a preprocessing step.</p> + * <p>Note that some conventions put the scalar part of the quaternion + * as the 4<sup>th</sup> component and the vector part as the first three + * components. This is <em>not</em> our convention. We put the scalar part + * as the first component.</p> + * @param q0 scalar part of the quaternion + * @param q1 first coordinate of the vectorial part of the quaternion + * @param q2 second coordinate of the vectorial part of the quaternion + * @param q3 third coordinate of the vectorial part of the quaternion + * @param needsNormalization if true, the coordinates are considered + * not to be normalized, a normalization preprocessing step is performed + * before using them + */ + public Rotation(double q0, double q1, double q2, double q3, + boolean needsNormalization) { + + if (needsNormalization) { + // normalization preprocessing + double inv = 1.0 / FastMath.sqrt(q0 * q0 + q1 * q1 + q2 * q2 + q3 * q3); + q0 *= inv; + q1 *= inv; + q2 *= inv; + q3 *= inv; + } + + this.q0 = q0; + this.q1 = q1; + this.q2 = q2; + this.q3 = q3; + + } + + /** Build a rotation from an axis and an angle. + * <p>We use the convention that angles are oriented according to + * the effect of the rotation on vectors around the axis. That means + * that if (i, j, k) is a direct frame and if we first provide +k as + * the axis and π/2 as the angle to this constructor, and then + * {@link #applyTo(Vector3D) apply} the instance to +i, we will get + * +j.</p> + * <p>Another way to represent our convention is to say that a rotation + * of angle θ about the unit vector (x, y, z) is the same as the + * rotation build from quaternion components { cos(-θ/2), + * x * sin(-θ/2), y * sin(-θ/2), z * sin(-θ/2) }. + * Note the minus sign on the angle!</p> + * <p>On the one hand this convention is consistent with a vectorial + * perspective (moving vectors in fixed frames), on the other hand it + * is different from conventions with a frame perspective (fixed vectors + * viewed from different frames) like the ones used for example in spacecraft + * attitude community or in the graphics community.</p> + * @param axis axis around which to rotate + * @param angle rotation angle. + * @exception ArithmeticException if the axis norm is zero + */ + public Rotation(Vector3D axis, double angle) { + + double norm = axis.getNorm(); + if (norm == 0) { + throw MathRuntimeException.createArithmeticException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_AXIS); + } + + double halfAngle = -0.5 * angle; + double coeff = FastMath.sin(halfAngle) / norm; + + q0 = FastMath.cos (halfAngle); + q1 = coeff * axis.getX(); + q2 = coeff * axis.getY(); + q3 = coeff * axis.getZ(); + + } + + /** Build a rotation from a 3X3 matrix. + + * <p>Rotation matrices are orthogonal matrices, i.e. unit matrices + * (which are matrices for which m.m<sup>T</sup> = I) with real + * coefficients. The module of the determinant of unit matrices is + * 1, among the orthogonal 3X3 matrices, only the ones having a + * positive determinant (+1) are rotation matrices.</p> + * + * <p>When a rotation is defined by a matrix with truncated values + * (typically when it is extracted from a technical sheet where only + * four to five significant digits are available), the matrix is not + * orthogonal anymore. This constructor handles this case + * transparently by using a copy of the given matrix and applying a + * correction to the copy in order to perfect its orthogonality. If + * the Frobenius norm of the correction needed is above the given + * threshold, then the matrix is considered to be too far from a + * true rotation matrix and an exception is thrown.<p> + * + * @param m rotation matrix + * @param threshold convergence threshold for the iterative + * orthogonality correction (convergence is reached when the + * difference between two steps of the Frobenius norm of the + * correction is below this threshold) + * + * @exception NotARotationMatrixException if the matrix is not a 3X3 + * matrix, or if it cannot be transformed into an orthogonal matrix + * with the given threshold, or if the determinant of the resulting + * orthogonal matrix is negative + * + */ + public Rotation(double[][] m, double threshold) + throws NotARotationMatrixException { + + // dimension check + if ((m.length != 3) || (m[0].length != 3) || + (m[1].length != 3) || (m[2].length != 3)) { + throw new NotARotationMatrixException( + LocalizedFormats.ROTATION_MATRIX_DIMENSIONS, + m.length, m[0].length); + } + + // compute a "close" orthogonal matrix + double[][] ort = orthogonalizeMatrix(m, threshold); + + // check the sign of the determinant + double det = ort[0][0] * (ort[1][1] * ort[2][2] - ort[2][1] * ort[1][2]) - + ort[1][0] * (ort[0][1] * ort[2][2] - ort[2][1] * ort[0][2]) + + ort[2][0] * (ort[0][1] * ort[1][2] - ort[1][1] * ort[0][2]); + if (det < 0.0) { + throw new NotARotationMatrixException( + LocalizedFormats.CLOSEST_ORTHOGONAL_MATRIX_HAS_NEGATIVE_DETERMINANT, + det); + } + + // There are different ways to compute the quaternions elements + // from the matrix. They all involve computing one element from + // the diagonal of the matrix, and computing the three other ones + // using a formula involving a division by the first element, + // which unfortunately can be zero. Since the norm of the + // quaternion is 1, we know at least one element has an absolute + // value greater or equal to 0.5, so it is always possible to + // select the right formula and avoid division by zero and even + // numerical inaccuracy. Checking the elements in turn and using + // the first one greater than 0.45 is safe (this leads to a simple + // test since qi = 0.45 implies 4 qi^2 - 1 = -0.19) + double s = ort[0][0] + ort[1][1] + ort[2][2]; + if (s > -0.19) { + // compute q0 and deduce q1, q2 and q3 + q0 = 0.5 * FastMath.sqrt(s + 1.0); + double inv = 0.25 / q0; + q1 = inv * (ort[1][2] - ort[2][1]); + q2 = inv * (ort[2][0] - ort[0][2]); + q3 = inv * (ort[0][1] - ort[1][0]); + } else { + s = ort[0][0] - ort[1][1] - ort[2][2]; + if (s > -0.19) { + // compute q1 and deduce q0, q2 and q3 + q1 = 0.5 * FastMath.sqrt(s + 1.0); + double inv = 0.25 / q1; + q0 = inv * (ort[1][2] - ort[2][1]); + q2 = inv * (ort[0][1] + ort[1][0]); + q3 = inv * (ort[0][2] + ort[2][0]); + } else { + s = ort[1][1] - ort[0][0] - ort[2][2]; + if (s > -0.19) { + // compute q2 and deduce q0, q1 and q3 + q2 = 0.5 * FastMath.sqrt(s + 1.0); + double inv = 0.25 / q2; + q0 = inv * (ort[2][0] - ort[0][2]); + q1 = inv * (ort[0][1] + ort[1][0]); + q3 = inv * (ort[2][1] + ort[1][2]); + } else { + // compute q3 and deduce q0, q1 and q2 + s = ort[2][2] - ort[0][0] - ort[1][1]; + q3 = 0.5 * FastMath.sqrt(s + 1.0); + double inv = 0.25 / q3; + q0 = inv * (ort[0][1] - ort[1][0]); + q1 = inv * (ort[0][2] + ort[2][0]); + q2 = inv * (ort[2][1] + ort[1][2]); + } + } + } + + } + + /** Build the rotation that transforms a pair of vector into another pair. + + * <p>Except for possible scale factors, if the instance were applied to + * the pair (u<sub>1</sub>, u<sub>2</sub>) it will produce the pair + * (v<sub>1</sub>, v<sub>2</sub>).</p> + * + * <p>If the angular separation between u<sub>1</sub> and u<sub>2</sub> is + * not the same as the angular separation between v<sub>1</sub> and + * v<sub>2</sub>, then a corrected v'<sub>2</sub> will be used rather than + * v<sub>2</sub>, the corrected vector will be in the (v<sub>1</sub>, + * v<sub>2</sub>) plane.</p> + * + * @param u1 first vector of the origin pair + * @param u2 second vector of the origin pair + * @param v1 desired image of u1 by the rotation + * @param v2 desired image of u2 by the rotation + * @exception IllegalArgumentException if the norm of one of the vectors is zero + */ + public Rotation(Vector3D u1, Vector3D u2, Vector3D v1, Vector3D v2) { + + // norms computation + double u1u1 = Vector3D.dotProduct(u1, u1); + double u2u2 = Vector3D.dotProduct(u2, u2); + double v1v1 = Vector3D.dotProduct(v1, v1); + double v2v2 = Vector3D.dotProduct(v2, v2); + if ((u1u1 == 0) || (u2u2 == 0) || (v1v1 == 0) || (v2v2 == 0)) { + throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR); + } + + double u1x = u1.getX(); + double u1y = u1.getY(); + double u1z = u1.getZ(); + + double u2x = u2.getX(); + double u2y = u2.getY(); + double u2z = u2.getZ(); + + // normalize v1 in order to have (v1'|v1') = (u1|u1) + double coeff = FastMath.sqrt (u1u1 / v1v1); + double v1x = coeff * v1.getX(); + double v1y = coeff * v1.getY(); + double v1z = coeff * v1.getZ(); + v1 = new Vector3D(v1x, v1y, v1z); + + // adjust v2 in order to have (u1|u2) = (v1|v2) and (v2'|v2') = (u2|u2) + double u1u2 = Vector3D.dotProduct(u1, u2); + double v1v2 = Vector3D.dotProduct(v1, v2); + double coeffU = u1u2 / u1u1; + double coeffV = v1v2 / u1u1; + double beta = FastMath.sqrt((u2u2 - u1u2 * coeffU) / (v2v2 - v1v2 * coeffV)); + double alpha = coeffU - beta * coeffV; + double v2x = alpha * v1x + beta * v2.getX(); + double v2y = alpha * v1y + beta * v2.getY(); + double v2z = alpha * v1z + beta * v2.getZ(); + v2 = new Vector3D(v2x, v2y, v2z); + + // preliminary computation (we use explicit formulation instead + // of relying on the Vector3D class in order to avoid building lots + // of temporary objects) + Vector3D uRef = u1; + Vector3D vRef = v1; + double dx1 = v1x - u1.getX(); + double dy1 = v1y - u1.getY(); + double dz1 = v1z - u1.getZ(); + double dx2 = v2x - u2.getX(); + double dy2 = v2y - u2.getY(); + double dz2 = v2z - u2.getZ(); + Vector3D k = new Vector3D(dy1 * dz2 - dz1 * dy2, + dz1 * dx2 - dx1 * dz2, + dx1 * dy2 - dy1 * dx2); + double c = k.getX() * (u1y * u2z - u1z * u2y) + + k.getY() * (u1z * u2x - u1x * u2z) + + k.getZ() * (u1x * u2y - u1y * u2x); + + if (c == 0) { + // the (q1, q2, q3) vector is in the (u1, u2) plane + // we try other vectors + Vector3D u3 = Vector3D.crossProduct(u1, u2); + Vector3D v3 = Vector3D.crossProduct(v1, v2); + double u3x = u3.getX(); + double u3y = u3.getY(); + double u3z = u3.getZ(); + double v3x = v3.getX(); + double v3y = v3.getY(); + double v3z = v3.getZ(); + + double dx3 = v3x - u3x; + double dy3 = v3y - u3y; + double dz3 = v3z - u3z; + k = new Vector3D(dy1 * dz3 - dz1 * dy3, + dz1 * dx3 - dx1 * dz3, + dx1 * dy3 - dy1 * dx3); + c = k.getX() * (u1y * u3z - u1z * u3y) + + k.getY() * (u1z * u3x - u1x * u3z) + + k.getZ() * (u1x * u3y - u1y * u3x); + + if (c == 0) { + // the (q1, q2, q3) vector is aligned with u1: + // we try (u2, u3) and (v2, v3) + k = new Vector3D(dy2 * dz3 - dz2 * dy3, + dz2 * dx3 - dx2 * dz3, + dx2 * dy3 - dy2 * dx3); + c = k.getX() * (u2y * u3z - u2z * u3y) + + k.getY() * (u2z * u3x - u2x * u3z) + + k.getZ() * (u2x * u3y - u2y * u3x); + + if (c == 0) { + // the (q1, q2, q3) vector is aligned with everything + // this is really the identity rotation + q0 = 1.0; + q1 = 0.0; + q2 = 0.0; + q3 = 0.0; + return; + } + + // we will have to use u2 and v2 to compute the scalar part + uRef = u2; + vRef = v2; + + } + + } + + // compute the vectorial part + c = FastMath.sqrt(c); + double inv = 1.0 / (c + c); + q1 = inv * k.getX(); + q2 = inv * k.getY(); + q3 = inv * k.getZ(); + + // compute the scalar part + k = new Vector3D(uRef.getY() * q3 - uRef.getZ() * q2, + uRef.getZ() * q1 - uRef.getX() * q3, + uRef.getX() * q2 - uRef.getY() * q1); + c = Vector3D.dotProduct(k, k); + q0 = Vector3D.dotProduct(vRef, k) / (c + c); + + } + + /** Build one of the rotations that transform one vector into another one. + + * <p>Except for a possible scale factor, if the instance were + * applied to the vector u it will produce the vector v. There is an + * infinite number of such rotations, this constructor choose the + * one with the smallest associated angle (i.e. the one whose axis + * is orthogonal to the (u, v) plane). If u and v are colinear, an + * arbitrary rotation axis is chosen.</p> + * + * @param u origin vector + * @param v desired image of u by the rotation + * @exception IllegalArgumentException if the norm of one of the vectors is zero + */ + public Rotation(Vector3D u, Vector3D v) { + + double normProduct = u.getNorm() * v.getNorm(); + if (normProduct == 0) { + throw MathRuntimeException.createIllegalArgumentException(LocalizedFormats.ZERO_NORM_FOR_ROTATION_DEFINING_VECTOR); + } + + double dot = Vector3D.dotProduct(u, v); + + if (dot < ((2.0e-15 - 1.0) * normProduct)) { + // special case u = -v: we select a PI angle rotation around + // an arbitrary vector orthogonal to u + Vector3D w = u.orthogonal(); + q0 = 0.0; + q1 = -w.getX(); + q2 = -w.getY(); + q3 = -w.getZ(); + } else { + // general case: (u, v) defines a plane, we select + // the shortest possible rotation: axis orthogonal to this plane + q0 = FastMath.sqrt(0.5 * (1.0 + dot / normProduct)); + double coeff = 1.0 / (2.0 * q0 * normProduct); + q1 = coeff * (v.getY() * u.getZ() - v.getZ() * u.getY()); + q2 = coeff * (v.getZ() * u.getX() - v.getX() * u.getZ()); + q3 = coeff * (v.getX() * u.getY() - v.getY() * u.getX()); + } + + } + + /** Build a rotation from three Cardan or Euler elementary rotations. + + * <p>Cardan rotations are three successive rotations around the + * canonical axes X, Y and Z, each axis being used once. There are + * 6 such sets of rotations (XYZ, XZY, YXZ, YZX, ZXY and ZYX). Euler + * rotations are three successive rotations around the canonical + * axes X, Y and Z, the first and last rotations being around the + * same axis. There are 6 such sets of rotations (XYX, XZX, YXY, + * YZY, ZXZ and ZYZ), the most popular one being ZXZ.</p> + * <p>Beware that many people routinely use the term Euler angles even + * for what really are Cardan angles (this confusion is especially + * widespread in the aerospace business where Roll, Pitch and Yaw angles + * are often wrongly tagged as Euler angles).</p> + * + * @param order order of rotations to use + * @param alpha1 angle of the first elementary rotation + * @param alpha2 angle of the second elementary rotation + * @param alpha3 angle of the third elementary rotation + */ + public Rotation(RotationOrder order, + double alpha1, double alpha2, double alpha3) { + Rotation r1 = new Rotation(order.getA1(), alpha1); + Rotation r2 = new Rotation(order.getA2(), alpha2); + Rotation r3 = new Rotation(order.getA3(), alpha3); + Rotation composed = r1.applyTo(r2.applyTo(r3)); + q0 = composed.q0; + q1 = composed.q1; + q2 = composed.q2; + q3 = composed.q3; + } + + /** Revert a rotation. + * Build a rotation which reverse the effect of another + * rotation. This means that if r(u) = v, then r.revert(v) = u. The + * instance is not changed. + * @return a new rotation whose effect is the reverse of the effect + * of the instance + */ + public Rotation revert() { + return new Rotation(-q0, q1, q2, q3, false); + } + + /** Get the scalar coordinate of the quaternion. + * @return scalar coordinate of the quaternion + */ + public double getQ0() { + return q0; + } + + /** Get the first coordinate of the vectorial part of the quaternion. + * @return first coordinate of the vectorial part of the quaternion + */ + public double getQ1() { + return q1; + } + + /** Get the second coordinate of the vectorial part of the quaternion. + * @return second coordinate of the vectorial part of the quaternion + */ + public double getQ2() { + return q2; + } + + /** Get the third coordinate of the vectorial part of the quaternion. + * @return third coordinate of the vectorial part of the quaternion + */ + public double getQ3() { + return q3; + } + + /** Get the normalized axis of the rotation. + * @return normalized axis of the rotation + * @see #Rotation(Vector3D, double) + */ + public Vector3D getAxis() { + double squaredSine = q1 * q1 + q2 * q2 + q3 * q3; + if (squaredSine == 0) { + return new Vector3D(1, 0, 0); + } else if (q0 < 0) { + double inverse = 1 / FastMath.sqrt(squaredSine); + return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse); + } + double inverse = -1 / FastMath.sqrt(squaredSine); + return new Vector3D(q1 * inverse, q2 * inverse, q3 * inverse); + } + + /** Get the angle of the rotation. + * @return angle of the rotation (between 0 and π) + * @see #Rotation(Vector3D, double) + */ + public double getAngle() { + if ((q0 < -0.1) || (q0 > 0.1)) { + return 2 * FastMath.asin(FastMath.sqrt(q1 * q1 + q2 * q2 + q3 * q3)); + } else if (q0 < 0) { + return 2 * FastMath.acos(-q0); + } + return 2 * FastMath.acos(q0); + } + + /** Get the Cardan or Euler angles corresponding to the instance. + + * <p>The equations show that each rotation can be defined by two + * different values of the Cardan or Euler angles set. For example + * if Cardan angles are used, the rotation defined by the angles + * a<sub>1</sub>, a<sub>2</sub> and a<sub>3</sub> is the same as + * the rotation defined by the angles π + a<sub>1</sub>, π + * - a<sub>2</sub> and π + a<sub>3</sub>. This method implements + * the following arbitrary choices:</p> + * <ul> + * <li>for Cardan angles, the chosen set is the one for which the + * second angle is between -π/2 and π/2 (i.e its cosine is + * positive),</li> + * <li>for Euler angles, the chosen set is the one for which the + * second angle is between 0 and π (i.e its sine is positive).</li> + * </ul> + * + * <p>Cardan and Euler angle have a very disappointing drawback: all + * of them have singularities. This means that if the instance is + * too close to the singularities corresponding to the given + * rotation order, it will be impossible to retrieve the angles. For + * Cardan angles, this is often called gimbal lock. There is + * <em>nothing</em> to do to prevent this, it is an intrinsic problem + * with Cardan and Euler representation (but not a problem with the + * rotation itself, which is perfectly well defined). For Cardan + * angles, singularities occur when the second angle is close to + * -π/2 or +π/2, for Euler angle singularities occur when the + * second angle is close to 0 or π, this implies that the identity + * rotation is always singular for Euler angles!</p> + * + * @param order rotation order to use + * @return an array of three angles, in the order specified by the set + * @exception CardanEulerSingularityException if the rotation is + * singular with respect to the angles set specified + */ + public double[] getAngles(RotationOrder order) + throws CardanEulerSingularityException { + + if (order == RotationOrder.XYZ) { + + // r (Vector3D.plusK) coordinates are : + // sin (theta), -cos (theta) sin (phi), cos (theta) cos (phi) + // (-r) (Vector3D.plusI) coordinates are : + // cos (psi) cos (theta), -sin (psi) cos (theta), sin (theta) + // and we can choose to have theta in the interval [-PI/2 ; +PI/2] + Vector3D v1 = applyTo(Vector3D.PLUS_K); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_I); + if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return new double[] { + FastMath.atan2(-(v1.getY()), v1.getZ()), + FastMath.asin(v2.getZ()), + FastMath.atan2(-(v2.getY()), v2.getX()) + }; + + } else if (order == RotationOrder.XZY) { + + // r (Vector3D.plusJ) coordinates are : + // -sin (psi), cos (psi) cos (phi), cos (psi) sin (phi) + // (-r) (Vector3D.plusI) coordinates are : + // cos (theta) cos (psi), -sin (psi), sin (theta) cos (psi) + // and we can choose to have psi in the interval [-PI/2 ; +PI/2] + Vector3D v1 = applyTo(Vector3D.PLUS_J); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_I); + if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return new double[] { + FastMath.atan2(v1.getZ(), v1.getY()), + -FastMath.asin(v2.getY()), + FastMath.atan2(v2.getZ(), v2.getX()) + }; + + } else if (order == RotationOrder.YXZ) { + + // r (Vector3D.plusK) coordinates are : + // cos (phi) sin (theta), -sin (phi), cos (phi) cos (theta) + // (-r) (Vector3D.plusJ) coordinates are : + // sin (psi) cos (phi), cos (psi) cos (phi), -sin (phi) + // and we can choose to have phi in the interval [-PI/2 ; +PI/2] + Vector3D v1 = applyTo(Vector3D.PLUS_K); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_J); + if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return new double[] { + FastMath.atan2(v1.getX(), v1.getZ()), + -FastMath.asin(v2.getZ()), + FastMath.atan2(v2.getX(), v2.getY()) + }; + + } else if (order == RotationOrder.YZX) { + + // r (Vector3D.plusI) coordinates are : + // cos (psi) cos (theta), sin (psi), -cos (psi) sin (theta) + // (-r) (Vector3D.plusJ) coordinates are : + // sin (psi), cos (phi) cos (psi), -sin (phi) cos (psi) + // and we can choose to have psi in the interval [-PI/2 ; +PI/2] + Vector3D v1 = applyTo(Vector3D.PLUS_I); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_J); + if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return new double[] { + FastMath.atan2(-(v1.getZ()), v1.getX()), + FastMath.asin(v2.getX()), + FastMath.atan2(-(v2.getZ()), v2.getY()) + }; + + } else if (order == RotationOrder.ZXY) { + + // r (Vector3D.plusJ) coordinates are : + // -cos (phi) sin (psi), cos (phi) cos (psi), sin (phi) + // (-r) (Vector3D.plusK) coordinates are : + // -sin (theta) cos (phi), sin (phi), cos (theta) cos (phi) + // and we can choose to have phi in the interval [-PI/2 ; +PI/2] + Vector3D v1 = applyTo(Vector3D.PLUS_J); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_K); + if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return new double[] { + FastMath.atan2(-(v1.getX()), v1.getY()), + FastMath.asin(v2.getY()), + FastMath.atan2(-(v2.getX()), v2.getZ()) + }; + + } else if (order == RotationOrder.ZYX) { + + // r (Vector3D.plusI) coordinates are : + // cos (theta) cos (psi), cos (theta) sin (psi), -sin (theta) + // (-r) (Vector3D.plusK) coordinates are : + // -sin (theta), sin (phi) cos (theta), cos (phi) cos (theta) + // and we can choose to have theta in the interval [-PI/2 ; +PI/2] + Vector3D v1 = applyTo(Vector3D.PLUS_I); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_K); + if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) { + throw new CardanEulerSingularityException(true); + } + return new double[] { + FastMath.atan2(v1.getY(), v1.getX()), + -FastMath.asin(v2.getX()), + FastMath.atan2(v2.getY(), v2.getZ()) + }; + + } else if (order == RotationOrder.XYX) { + + // r (Vector3D.plusI) coordinates are : + // cos (theta), sin (phi1) sin (theta), -cos (phi1) sin (theta) + // (-r) (Vector3D.plusI) coordinates are : + // cos (theta), sin (theta) sin (phi2), sin (theta) cos (phi2) + // and we can choose to have theta in the interval [0 ; PI] + Vector3D v1 = applyTo(Vector3D.PLUS_I); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_I); + if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return new double[] { + FastMath.atan2(v1.getY(), -v1.getZ()), + FastMath.acos(v2.getX()), + FastMath.atan2(v2.getY(), v2.getZ()) + }; + + } else if (order == RotationOrder.XZX) { + + // r (Vector3D.plusI) coordinates are : + // cos (psi), cos (phi1) sin (psi), sin (phi1) sin (psi) + // (-r) (Vector3D.plusI) coordinates are : + // cos (psi), -sin (psi) cos (phi2), sin (psi) sin (phi2) + // and we can choose to have psi in the interval [0 ; PI] + Vector3D v1 = applyTo(Vector3D.PLUS_I); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_I); + if ((v2.getX() < -0.9999999999) || (v2.getX() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return new double[] { + FastMath.atan2(v1.getZ(), v1.getY()), + FastMath.acos(v2.getX()), + FastMath.atan2(v2.getZ(), -v2.getY()) + }; + + } else if (order == RotationOrder.YXY) { + + // r (Vector3D.plusJ) coordinates are : + // sin (theta1) sin (phi), cos (phi), cos (theta1) sin (phi) + // (-r) (Vector3D.plusJ) coordinates are : + // sin (phi) sin (theta2), cos (phi), -sin (phi) cos (theta2) + // and we can choose to have phi in the interval [0 ; PI] + Vector3D v1 = applyTo(Vector3D.PLUS_J); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_J); + if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return new double[] { + FastMath.atan2(v1.getX(), v1.getZ()), + FastMath.acos(v2.getY()), + FastMath.atan2(v2.getX(), -v2.getZ()) + }; + + } else if (order == RotationOrder.YZY) { + + // r (Vector3D.plusJ) coordinates are : + // -cos (theta1) sin (psi), cos (psi), sin (theta1) sin (psi) + // (-r) (Vector3D.plusJ) coordinates are : + // sin (psi) cos (theta2), cos (psi), sin (psi) sin (theta2) + // and we can choose to have psi in the interval [0 ; PI] + Vector3D v1 = applyTo(Vector3D.PLUS_J); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_J); + if ((v2.getY() < -0.9999999999) || (v2.getY() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return new double[] { + FastMath.atan2(v1.getZ(), -v1.getX()), + FastMath.acos(v2.getY()), + FastMath.atan2(v2.getZ(), v2.getX()) + }; + + } else if (order == RotationOrder.ZXZ) { + + // r (Vector3D.plusK) coordinates are : + // sin (psi1) sin (phi), -cos (psi1) sin (phi), cos (phi) + // (-r) (Vector3D.plusK) coordinates are : + // sin (phi) sin (psi2), sin (phi) cos (psi2), cos (phi) + // and we can choose to have phi in the interval [0 ; PI] + Vector3D v1 = applyTo(Vector3D.PLUS_K); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_K); + if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return new double[] { + FastMath.atan2(v1.getX(), -v1.getY()), + FastMath.acos(v2.getZ()), + FastMath.atan2(v2.getX(), v2.getY()) + }; + + } else { // last possibility is ZYZ + + // r (Vector3D.plusK) coordinates are : + // cos (psi1) sin (theta), sin (psi1) sin (theta), cos (theta) + // (-r) (Vector3D.plusK) coordinates are : + // -sin (theta) cos (psi2), sin (theta) sin (psi2), cos (theta) + // and we can choose to have theta in the interval [0 ; PI] + Vector3D v1 = applyTo(Vector3D.PLUS_K); + Vector3D v2 = applyInverseTo(Vector3D.PLUS_K); + if ((v2.getZ() < -0.9999999999) || (v2.getZ() > 0.9999999999)) { + throw new CardanEulerSingularityException(false); + } + return new double[] { + FastMath.atan2(v1.getY(), v1.getX()), + FastMath.acos(v2.getZ()), + FastMath.atan2(v2.getY(), -v2.getX()) + }; + + } + + } + + /** Get the 3X3 matrix corresponding to the instance + * @return the matrix corresponding to the instance + */ + public double[][] getMatrix() { + + // products + double q0q0 = q0 * q0; + double q0q1 = q0 * q1; + double q0q2 = q0 * q2; + double q0q3 = q0 * q3; + double q1q1 = q1 * q1; + double q1q2 = q1 * q2; + double q1q3 = q1 * q3; + double q2q2 = q2 * q2; + double q2q3 = q2 * q3; + double q3q3 = q3 * q3; + + // create the matrix + double[][] m = new double[3][]; + m[0] = new double[3]; + m[1] = new double[3]; + m[2] = new double[3]; + + m [0][0] = 2.0 * (q0q0 + q1q1) - 1.0; + m [1][0] = 2.0 * (q1q2 - q0q3); + m [2][0] = 2.0 * (q1q3 + q0q2); + + m [0][1] = 2.0 * (q1q2 + q0q3); + m [1][1] = 2.0 * (q0q0 + q2q2) - 1.0; + m [2][1] = 2.0 * (q2q3 - q0q1); + + m [0][2] = 2.0 * (q1q3 - q0q2); + m [1][2] = 2.0 * (q2q3 + q0q1); + m [2][2] = 2.0 * (q0q0 + q3q3) - 1.0; + + return m; + + } + + /** Apply the rotation to a vector. + * @param u vector to apply the rotation to + * @return a new vector which is the image of u by the rotation + */ + public Vector3D applyTo(Vector3D u) { + + double x = u.getX(); + double y = u.getY(); + double z = u.getZ(); + + double s = q1 * x + q2 * y + q3 * z; + + return new Vector3D(2 * (q0 * (x * q0 - (q2 * z - q3 * y)) + s * q1) - x, + 2 * (q0 * (y * q0 - (q3 * x - q1 * z)) + s * q2) - y, + 2 * (q0 * (z * q0 - (q1 * y - q2 * x)) + s * q3) - z); + + } + + /** Apply the inverse of the rotation to a vector. + * @param u vector to apply the inverse of the rotation to + * @return a new vector which such that u is its image by the rotation + */ + public Vector3D applyInverseTo(Vector3D u) { + + double x = u.getX(); + double y = u.getY(); + double z = u.getZ(); + + double s = q1 * x + q2 * y + q3 * z; + double m0 = -q0; + + return new Vector3D(2 * (m0 * (x * m0 - (q2 * z - q3 * y)) + s * q1) - x, + 2 * (m0 * (y * m0 - (q3 * x - q1 * z)) + s * q2) - y, + 2 * (m0 * (z * m0 - (q1 * y - q2 * x)) + s * q3) - z); + + } + + /** Apply the instance to another rotation. + * Applying the instance to a rotation is computing the composition + * in an order compliant with the following rule : let u be any + * vector and v its image by r (i.e. r.applyTo(u) = v), let w be the image + * of v by the instance (i.e. applyTo(v) = w), then w = comp.applyTo(u), + * where comp = applyTo(r). + * @param r rotation to apply the rotation to + * @return a new rotation which is the composition of r by the instance + */ + public Rotation applyTo(Rotation r) { + return new Rotation(r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3), + r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2), + r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3), + r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1), + false); + } + + /** Apply the inverse of the instance to another rotation. + * Applying the inverse of the instance to a rotation is computing + * the composition in an order compliant with the following rule : + * let u be any vector and v its image by r (i.e. r.applyTo(u) = v), + * let w be the inverse image of v by the instance + * (i.e. applyInverseTo(v) = w), then w = comp.applyTo(u), where + * comp = applyInverseTo(r). + * @param r rotation to apply the rotation to + * @return a new rotation which is the composition of r by the inverse + * of the instance + */ + public Rotation applyInverseTo(Rotation r) { + return new Rotation(-r.q0 * q0 - (r.q1 * q1 + r.q2 * q2 + r.q3 * q3), + -r.q1 * q0 + r.q0 * q1 + (r.q2 * q3 - r.q3 * q2), + -r.q2 * q0 + r.q0 * q2 + (r.q3 * q1 - r.q1 * q3), + -r.q3 * q0 + r.q0 * q3 + (r.q1 * q2 - r.q2 * q1), + false); + } + + /** Perfect orthogonality on a 3X3 matrix. + * @param m initial matrix (not exactly orthogonal) + * @param threshold convergence threshold for the iterative + * orthogonality correction (convergence is reached when the + * difference between two steps of the Frobenius norm of the + * correction is below this threshold) + * @return an orthogonal matrix close to m + * @exception NotARotationMatrixException if the matrix cannot be + * orthogonalized with the given threshold after 10 iterations + */ + private double[][] orthogonalizeMatrix(double[][] m, double threshold) + throws NotARotationMatrixException { + double[] m0 = m[0]; + double[] m1 = m[1]; + double[] m2 = m[2]; + double x00 = m0[0]; + double x01 = m0[1]; + double x02 = m0[2]; + double x10 = m1[0]; + double x11 = m1[1]; + double x12 = m1[2]; + double x20 = m2[0]; + double x21 = m2[1]; + double x22 = m2[2]; + double fn = 0; + double fn1; + + double[][] o = new double[3][3]; + double[] o0 = o[0]; + double[] o1 = o[1]; + double[] o2 = o[2]; + + // iterative correction: Xn+1 = Xn - 0.5 * (Xn.Mt.Xn - M) + int i = 0; + while (++i < 11) { + + // Mt.Xn + double mx00 = m0[0] * x00 + m1[0] * x10 + m2[0] * x20; + double mx10 = m0[1] * x00 + m1[1] * x10 + m2[1] * x20; + double mx20 = m0[2] * x00 + m1[2] * x10 + m2[2] * x20; + double mx01 = m0[0] * x01 + m1[0] * x11 + m2[0] * x21; + double mx11 = m0[1] * x01 + m1[1] * x11 + m2[1] * x21; + double mx21 = m0[2] * x01 + m1[2] * x11 + m2[2] * x21; + double mx02 = m0[0] * x02 + m1[0] * x12 + m2[0] * x22; + double mx12 = m0[1] * x02 + m1[1] * x12 + m2[1] * x22; + double mx22 = m0[2] * x02 + m1[2] * x12 + m2[2] * x22; + + // Xn+1 + o0[0] = x00 - 0.5 * (x00 * mx00 + x01 * mx10 + x02 * mx20 - m0[0]); + o0[1] = x01 - 0.5 * (x00 * mx01 + x01 * mx11 + x02 * mx21 - m0[1]); + o0[2] = x02 - 0.5 * (x00 * mx02 + x01 * mx12 + x02 * mx22 - m0[2]); + o1[0] = x10 - 0.5 * (x10 * mx00 + x11 * mx10 + x12 * mx20 - m1[0]); + o1[1] = x11 - 0.5 * (x10 * mx01 + x11 * mx11 + x12 * mx21 - m1[1]); + o1[2] = x12 - 0.5 * (x10 * mx02 + x11 * mx12 + x12 * mx22 - m1[2]); + o2[0] = x20 - 0.5 * (x20 * mx00 + x21 * mx10 + x22 * mx20 - m2[0]); + o2[1] = x21 - 0.5 * (x20 * mx01 + x21 * mx11 + x22 * mx21 - m2[1]); + o2[2] = x22 - 0.5 * (x20 * mx02 + x21 * mx12 + x22 * mx22 - m2[2]); + + // correction on each elements + double corr00 = o0[0] - m0[0]; + double corr01 = o0[1] - m0[1]; + double corr02 = o0[2] - m0[2]; + double corr10 = o1[0] - m1[0]; + double corr11 = o1[1] - m1[1]; + double corr12 = o1[2] - m1[2]; + double corr20 = o2[0] - m2[0]; + double corr21 = o2[1] - m2[1]; + double corr22 = o2[2] - m2[2]; + + // Frobenius norm of the correction + fn1 = corr00 * corr00 + corr01 * corr01 + corr02 * corr02 + + corr10 * corr10 + corr11 * corr11 + corr12 * corr12 + + corr20 * corr20 + corr21 * corr21 + corr22 * corr22; + + // convergence test + if (FastMath.abs(fn1 - fn) <= threshold) + return o; + + // prepare next iteration + x00 = o0[0]; + x01 = o0[1]; + x02 = o0[2]; + x10 = o1[0]; + x11 = o1[1]; + x12 = o1[2]; + x20 = o2[0]; + x21 = o2[1]; + x22 = o2[2]; + fn = fn1; + + } + + // the algorithm did not converge after 10 iterations + throw new NotARotationMatrixException( + LocalizedFormats.UNABLE_TO_ORTHOGONOLIZE_MATRIX, + i - 1); + } + + /** Compute the <i>distance</i> between two rotations. + * <p>The <i>distance</i> is intended here as a way to check if two + * rotations are almost similar (i.e. they transform vectors the same way) + * or very different. It is mathematically defined as the angle of + * the rotation r that prepended to one of the rotations gives the other + * one:</p> + * <pre> + * r<sub>1</sub>(r) = r<sub>2</sub> + * </pre> + * <p>This distance is an angle between 0 and π. Its value is the smallest + * possible upper bound of the angle in radians between r<sub>1</sub>(v) + * and r<sub>2</sub>(v) for all possible vectors v. This upper bound is + * reached for some v. The distance is equal to 0 if and only if the two + * rotations are identical.</p> + * <p>Comparing two rotations should always be done using this value rather + * than for example comparing the components of the quaternions. It is much + * more stable, and has a geometric meaning. Also comparing quaternions + * components is error prone since for example quaternions (0.36, 0.48, -0.48, -0.64) + * and (-0.36, -0.48, 0.48, 0.64) represent exactly the same rotation despite + * their components are different (they are exact opposites).</p> + * @param r1 first rotation + * @param r2 second rotation + * @return <i>distance</i> between r1 and r2 + */ + public static double distance(Rotation r1, Rotation r2) { + return r1.applyInverseTo(r2).getAngle(); + } + +} |